=Paper=
{{Paper
|id=Vol-2076/paper-12
|storemode=property
|title=Mathematical Modeling of the Autodyne Signal Characteristics at Strong Reflected Emission
|pdfUrl=https://ceur-ws.org/Vol-2076/paper-12.pdf
|volume=Vol-2076
|authors=Vladislav Ya. Noskov,Kirill A. Ignatkov,Andrey P. Chupakhin
}}
==Mathematical Modeling of the Autodyne Signal Characteristics at Strong Reflected Emission==
https://ceur-ws.org/Vol-2076/paper-12.pdf
Mathematical Modeling of the Autodyne Signal
Characteristics at Strong Reflected Emission
Vladislav Ya. Noskov, Kirill A. Ignatkov, and Andrey P. Chupakhin
Ural Federal University named after the First President of Russia B.N.Yeltsin,
The Department of radioelectronics and information technology,
Yekaterinburg, Russia
noskov@oko-ek.ru,
Abstract. On the base of developed mathematical model of microwave
oscillator interaction with the strong reflected emission, with attraction
of numerical methods, the research results are presented for the features
of the autodyne signal characteristics formation. Researches are fulfilled
for the microwave oscillator model with a single-circuit oscillating system,
taking into account its non-isochronity and non-isodromity. An influence
of inherent parameters and the reflected emission delay phenomenon onto
the autodune response formation is revealed. Graphs of normalized signal
charcteristics are obtained showing the autodyne response shape, and its
spectral analysis is performed. Harmonic coefficients and amplitudes of
spectrum harmonic components are calculated together with averaged
value levels of the autodyne response as a function of reflection coefficient
modulus at various distance of the radar object.
Keywords: autodyne, microwave oscillator, strong reflected emission,
autodyne response, mathematical modeling of signals.
1 Introduction
Research results of autodyne microwave oscillator (MO) at strong reflected emis-
sion represent the practical interest in various areas of science and engineering.
They are claimed at determination of output signal formation features, at es-
timation of the dynamic range, for analysis of parameter measurement errors
and for a choice of the optimal operation mode of the short-range radar sensors,
which use the autodyne principle of the transceiver architecture.
Specifics of these sensor functioning at problem solution, for instance, de-
termination of electric-physical and dynamic parameters of the radar objects
consists in the fact that the distance between a sensor and an object can be
extremely small (down to zero) under several conditions. At that, the reflected
emission level may turn out to be commensurable with the level of probing emis-
sion. In contrast to the enough studied case of the weak reflected emission, the
case of the MO impact of strong emission was investigated insufficiently. It is
examined in the known literature on the base of experimental data and modeling
results on the analog computers only [1, 2].
104
In this paper, on the base of developed mathematical model with involving
of numerical approaches, we present the analysis results of the single-circuit
autodyne, which partially fill a mentioned gap.
2 Mathematical model of the autodyne at strong
reflected emission
The functional block diagram of the radar sensor with the autodyne architecture
principle of the transceiver is presented in fig. 1. Electromagnetic oscillations pro-
duced by MO are emitted through the receiving-transmitting antenna towards
the radar object. Microwave emission reflected from the object returns through
the same antenna into MO and causes the autodyne effect.
Fig. 1. Functional block diagram of the autodyne MO
As we know [1], the autodyne effect consists in variations of amplitude and a
frequency of MO oscillations. At that, arisen autodyne variations of the current
in the power source circuit of the MO active element (AE) are transformed
into the auto-detection signal uad by means of the registration unit. In some
autodyne sensor constructions, the useful signal is obtained with the help of
external detection circuit ued ,which transforms the autodyne variations of an
amplitude or frequency of microwave oscillations to the output signal voltage.
The equivalent diagram of the autodyne MO can be represented in the form
of parallel connection of averaged (over the oscillation period) AE conductivity
YAE = YAE (A, ω), depended on the amplitude A and the current frequency ω
of oscillations, the oscillating system (OS) YOS (ω) and the load YL = YL (ω).
Oscillation equation for this circuit has a form [1, 3]:
YAE (A, ω) + YOS (ω) + YL (ω) . (1)
A difficulty of analytical solution finding of this equation consist in the pres-
ence of nonlinear dependences of all its terms upon oscillation parameters. At
that, we should note that the YL (ω) conductivity in (1) also depends on the
delay time τ of the emission, at that
YL (ω) ≡ YL (ω, τ ) = GL (ω, τ ) + jBL (ω, τ ) . (2)
105
Here
G0 (1 − Γ 2 )
GL (ω, τ ) = ,
1 + Γ 2 + 2Γ cos δ(ω, τ )
2G0 Γ sin δ(ω, τ )
BL (ω, τ ) = ;
1 + Γ 2 + 2Γ cos δ(ω, τ )
Γ is a modulus of the reflection coefficient reduced to MO terminals, which
characterizes the emission attenuation at its propagation to the radar object
and back; δ(ω, τ ) is the total phase incursion; G0 is the load conductivity at
reflected emission absence.
To simplify analysis of (1) without loss of generality, we take some assump-
tions. We shall limit the present research by the case of autodyne response ex-
traction of oscillation amplitude variation with the help of the external detector.
In addition, we shall examine the processes in MO in the form of variations of
oscillation parameters in the vicinity of its steady-state, when at Γ = 0 we have:
A = A0 , ω = ω0 For this, we represent the oscillation amplitude and frequency
in the form: A = A0 + ∆A, ω = ω0 + ∆ω where ∆A and ∆ω are appropriate
variations of the MO steady-state at Γ 6= 0.
Then, acting in accordance the accepted approach [1], from (1) with account
of (2) for the case of MO with single-circuit OS, we obtain the system of linearized
equations for determination of relative amplitude variations a = ∆A A0 and the
frequency of oscillation χ = ∆ωω0 :
Γ − cos δ(ω, τ )
αa + εχ + Γ η =0 ; (3)
1 + Γ 2 + 2Γ cos δ(ω, τ )
sin δ(ω, τ )
βa + QL χ + Γ η 2
=0 , (4)
1 + Γ + 2Γ cos δ(ω, τ )
where α,ε,β are dimensionless parameters determining the limit cycle strength,
non-isodromity and non-isochronity of MO, relatively [4]; η = GG0 is an efficiency
and Γ is the conduction of all OS losses.
Under real functioning conditions of the autodyne sensor, the high level of
reflected emission is observed on the extremely short distance to the radar object.
Under such autodyne operation conditions, it is really acceptable to assume:
δ(ω, τ ) = ωτ [5]. Then, the expression for the phase δ(ω, τ ) can be written in
the form:
δ(ω, τ ) = δ(χ, τn ) = 2π(1 + χ)(N + τn ) , (5)
where τn = ω0 τ is normalized time; N = 2l λ is the integer number of half-
wavelengths, which falls between a sensor and the radar object.
3 Calculation and analysis of signal characteristics
Main signal autodyne characteristics are functions of relative variations of the
amplitude a and the frequency χ of oscillations versus variations of delay time
106
τn of reflected emission [4, 5]. The first function is called the autodyne amplitude
characteristic (AAC), while the second function autodyne frequency character-
istic (AFC) [6]. For calculation of these characteristics, we rewrite (3) and (4)
with account of (5), assuming η = 1, as follows:
Γ ρ γBc (χ, τn ) − Bs (χ, τn ) Bc (χ, τn )
a(τn ) + + =0 ; (6)
α 1 − γρ ρ
γBc (χ, τn ) − Bs (χ, τn )
χ(τn ) − Γ =0 , (7)
QL (1 − γρ)
where
Γ − cos 2π(1 + χ)(N + τn )
Bc (χ, τn ) = ;
1 + Γ 2 + 2Γ cos 2π(1 + χ)(N + τn )
sin 2π(1 + χ)(N + τn )
Bs (χ, τn ) = ;
1 + Γ 2 + 2Γ cos 2π(1 + χ)(N + τn )
β
γ= α , γ = QεL are non-isochronity and non-isodromity coefficients of MO, QL
is the loaded Q - factor.
The solution of equations (6), (7) we find using mathematical packet Math-
CAD. For this, at first, we find the solution for autodyne frequency variations
χ of the transcendent equation (7) by the secant method o the iteration algo-
rithm realized in the root function. After substitution of obtained values of χ
into equation (6), we obtain values of a variable by the same approach.
c) d)
a) b)
Fig. 2. Plots of AFC χn (τn ) (a) and AAC an (τn ) (b) and their spectra χn (Fn ) (c) and
an (Fn ) (d) calculated at N = 1; γ = ρ = 0 and various values of Γ : Γ = 0.01 (curves
1) and Γ = 0.5 (curves 2)
Then, performing a search of local extreems in functions a = a(τn ) and
χ = χ(τn ) with the help of embedded Maximize (f, x1 , ...xm ) function, we obtain
maximal values of autodyne variation deviations amax and χmax . After that, we
perform normalization of a = a(τn ) and χ = χ(τn ) function with respect to
extreme values amax and χmax obtained. At the end, we obtain the required
signal characteristics in the normalized form: an = a(τ n) χ(τn )
amax and χn = χmax .
The developed calculation algorithm according to (6) and (7) was verified
for obtained results convergence at the small signal, when Γ < 1, with results of
107
signal characteristics calculations, which were obtained starting from the small-
signal analysis fulfilled in [5, 6].
At first, we perform the analysis for the case of the extremely short distance
to the radar object assuming N = 1. Then, we reveal the time delay τ influence
on features of autodyne response formation assuming N = 100. At that, for each
case, we introduce variations in inherent MO parameters fulfilling calculations,
at first, for isochronous and isodromous case (γ = ρ = 0), and then for non-
isochronous and non-isodromous autodyne MO, when γ 6= 0 and ρ 6= 0.
a) b)
Fig. 3. Plots of coefficients KAF C , KACC , the FB parameter CF B and the level of
spectral components χn (Fn ) (a) and an (Fn ) (b) (f orn = 1, 5) of the autodyne response
of the isochronous MO, as well as the average value of an (0) versus Γ calculated for
N = 1; γ = ρ = 0
a) b) c) d)
e) f)
Fig. 4. AFC and AAC (a)(d) and spectra (e), (f) of the autodyne response of non-
isochronous and non-isodromous MO calculated at N = 1 and various values of Γ :
Γ = 0.01 (curves 1), Γ = 0.5 (curves 2) and for following parameters: γ = 1.5; ρ = 0.1
(a), (b), (e) and γ = 1.5; ρ = 0.1 (c), (d), (f)
108
Figures 2 (a), (b) show AFC χn (τn ) and AAC an (τn ) in the form of time-
diagrams for the isochronous MO. Plots1 were calculated at various values of
Γ . For strong reflected emission, when Γ = 0.5, (see curves 2), we perform the
expansion of χn (τn ) and an (τn ) into the harmonic Fourier series while Figs. 2
(c), (d) show the corresponding spectra.
a) b)
Fig. 5. Plots of coefficients KAF C , KACC , the FB parameter CF B and the level of
spectral components χn (Fn ) (a) and an (Fn ) (b) (f orn = 1, 5) of the autodyne responce
of the non-isochronous and non-isodromous MO, as well as average valuesχn (0) and
an (0) (at n=0) versus Γ calculated for N = 1; γ = ±1.5; ρ = ±0.1
c) d)
a) b)
Fig. 6. Plots of AFC (a) and AAC (b) and spectra χn (Fn ) (c), an (Fn ) (d) of the
autodyne responce of the isochronious MO calculated at Γ = 0.1; γ = ρ = 0 and
N : N = 1 (curves 1) and N = 100 (curves 2)
Plots of harmonic coefficients KAF C and KACC , the level of harmonic com-
ponents of the autodyne response spectra on the frequency χn (F n) and of am-
plitude an (F n) variations, as well as the average value an (0) versus the reflection
coefficient modulus Γ of the isochronous MO are presented in Figs. 3(a) and (b).
Here we show plots of feedback (FB) parameter CF B defining as a product of
the delay time of reflected emission by the autodyne frequency deviation [4]. The
shape of these plots in Fig. 3 is broken at Γ = 0.7, for which the instantaneous
signal characteristic value jumps begin.
1
Hereafter, we take the following values: QL = 100; α = 0.1.
109
a) b)
Fig. 7. Plots of coefficients KAF C , KACC , the FB parameter CF B and the level of
spectral components χn (Fn ) (a) and amplitudes an (Fn ) (b) (at n = 1, ...5) of spectra
and average value (at n = 0) of the isocronouse MO versus the Γ coefficient calculated
for N = 100; γ = ρ = 0
For positive (γ = 1.5; ρ = 0.1) and negative (γ = −1.5; ρ = −0.1) coefficients
of non-isochronity and non-isodromity of MO, Figs. 4(a) (d) show AFC χn (τn )
and AAC an (τn ) which are calculated at the previous value of distance (N = 1),
but at different values of the reflection coefficient Γ : Γ = 0.01 (curves 1) and
Γ = 0.5 (curves 2).
Spectra χn (Fn ) and an (Fn ) are presented in Figs. 4(e) and (f). Functions
KAF C (Γ ), KACC (Γ ) and CF B (Γ ) are shown in Fig. 5 (a) and (b). Here, we
present curves of the harmonic components level χn (Fn ), an (Fn ) (at n = 1, ...5)
and χn (0), an (0) (n = 0) versus Γ coefficient. The shape of all these plots, as
we see from Fig. 5, are broken at Γ = 0.5.
c) d)
a) b)
Fig. 8. Plots of AFC (a) and AAC (b) and spectra χn (Fn ) (c), an (Fn ) (d) of the
autodyne responce of non-isochronious MO calculated at Γ = 0.7; γ = 1.5; ρ = 0.1
and various N : N = 1 (curves 1) and N = 100 (curves 2)
The influence of the radar object on the shape and the spectrum of the
autodyne signal for the case of isochronous MO is presented on plots in Fig. 6.
We observe AFC χn (τn ) (a), AAC an (τn ) (b) and their spectra (c), (d) calculated
at Γ = 0.1 and various values of the half-wavelengths number N from the sensor
to the radar object: N = 1 (curves 1) and N = 100 (curves 2). Figures 7(a) and
(b) show the plots of harmonic coefficients KAF C and KACC , the FB parameter
CF B and the level of spectra harmonic components χn (Fn ) (a) and an (Fn ) (b)
110
(when n = 1, ...5), as well as average values of the autodyne response (at n = 0)
versus of the reflection coefficient modulus Γ .
a) b)
Fig. 9. Plots of coefficients KAF C , KACC , the FB parameter CF B and the level of
spectral components χn (Fn ) (a) and amplitudes an (Fn ) (b) (at n = 1, ...5) of spec-
tra and average value (at n = 0) of thenon-isocronouse MO versus the Γ coefficient
calculated for N = 100; γ = 1.5; ρ = 0.1
For the case of non-isochronous and non-isodronous MO AFC χn (τn ), AAC
an (τn ), calculated at various number of half-waves: N = 1 (curves 1) and N =
100 (curves 2), are presented in plots in Fig. 8 (a) and (b). Spectra calculation
results for the case N = 100, are presented in Figs. 8(c) and (d). Functions
KAF C , KACC , CF B , as well as levels χn (Fn ), an (Fn ) (n = 1, ...5), and χn (0),
an (0) at(n = 0) versus Γ coefficient are presented in Figs. 9(a) and (b). As we
see from Fig. 9, in this case plots are broken at Γ =0.08.
4 Conclusions
The resume of analysis of numerical modeling of the autodyne response formation
processes in MO obtained results consist in the following.
Plots of normalized functions of reactive bn (τn ) and resistive gn (τn ) compo-
nents of the MO load conduction versus the normalized time τn are presented in
Figs. 2(a) and (b) of the paper [7]. From comparison of these plots with obtained
by us curves of AFC χn (τn ) and AAC an (τn ) of the isochronous MO (see Figs.
2(a) and (b)), we see that the last ones practically repeat the former, but with
inversion of instantaneous values.
Spectra of reactive bn (Fn ) and active gn (Fn ) load conduction (see Figs. 3(c),
(d) of the paper [7]) are similar to the appropriate spectra χn (Fn ) and an (Fn ),
which are presented in Figs. 2(c), (d) of the present paper. In addition, as we see
from plots comparison in Fig. 4 of the paper [7] and presented by us in Fig. 3, the
shape of harmonic coefficients Kb and KAF C , Kg and KACC practically coin-
cides. At that, the relative values of harmonic levels bn (n) and χn (n), gn (n) and
an (n) and average values gn (0) and an (0) are also in the qualitative agreement.
Characteristic comparison results presented here allow conclusion that au-
todyne response distortions in the case of strong reflected emission are caused
111
predominantly by the action of the load nonlinearity rather than the signal re-
striction by the AE electronic conductance, as was assumed in [1]. We must also
note that in the case of isochronous MO, the constant component in χn (τn )
autodyne characteristic is absent, as for bn (χn ) component of the load conduc-
tance. Signal jumps, as it was mentioned earlier, begin at Γ = 0.7, at that, the
FB parameter CF B = 0.35 (see Figs. 3(a), (b)).
MO non-isochronity, as can be seen from comparison of AFC χn (τn ) and
AAC an (τn ) in Figs. 2(a), (b) and Figs. 4(a)-(d), causes the phase offset of
χn (τn ) characteristics by the angle θ = tan−1 (γ), which for chosen parameters
(γ = ±1.5) for the positive value of γ, is θ ≈ 1, while for the negative values
θ1. MO non-isodromity causes the AAC phase offset by the angle ψ = tan−1 (ρ).
Since QL1 >> 1, the angle ψ is comparatively small in value and usually does
not exceed ±0.5. In here considered case, its value is ψ ≈ 0.1 at ρ = 0.1 (see
Fig. 4(b)) and ψ ≈ 0.1 at ρ = 0.1 (see Fig. 4(d)). These phase offsets, as wee
see from the curves in Figs. 4(a) (d), cause additional clutter of the autodyne
response, which are expressed in appearance of characteristic wave tilts to one
or another side depending on the ratio of magnitudes and signs of coefficients of
non-isochronity γ and non-isodromity ρ of MO.
Besides, MO non-isochronity in the case of strong reflected emission causes
on AFC the increase of frequency deviation and appearance of DC component,
the sign and magnitude of which depend on the sign and the magnitude of γ
coefficient. The first phenomenon leads to increase the autodyne CF B parameter,
while the second one lead to the offset of the central frequency of oscillation.
Therefore, during increase of the reflection emission level of the isochronous
MO, the appearance of signal jumps (see Figs. 5(a), (b)) is observed at lesser
reflection coefficient (Γ = 0.5), than for the isochronous MO (Γ = 0.7) (see
Figs. 3(a), (b)). We should also note that the amplitude spectrum picture for
sign change of non-isochronity coefficient γ does not practically change (see Figs.
4(e), (f)). This means that for sign change of γ or movement direction change
of the radar object, variations of phase relations occur for harmonic components
in the autodyne response spectrum.
From analysis of above cases, it follows that for strong reflected emission,
when the reflection coeffiient Γ in commensurable with one, jumps in the process
of signal formation begin not for CF B = 1, as in the case of weak signals [1, 6],
but for its lesser values. So, for the case of the isochronous oscillator, jumps
begin at CF B = 0.35 (see Fig. 3), while for non-isochronous at CF B = 0.7 (see
Fig. 5).
The increase of radar object distance causes the appropriate growth of the
CF B parameter and appearance of signal jumps at lower level of reflected emis-
sion. Therefore, at studying of the distance variation influence (the number of
half-wavelengths N) on features of the autodyne response formation in the mode
of strong reflected emission, the range of analyzed values of the reflection co-
efficient Γ are proportionally narrowed and passes towards the area of weak
signals.
112
Such a situation found its interpretation in values of Γ chosen for calculations
on plots in Figs. 6-9. From these plots, we also see that the presence of the DC
component in autodyne frequency variations of the isochronous is absent as
in the case of extremely small values of N , whereas for the non-isochronous
MO this dependence happens at radar object distance increase. There are no
other qualitative differences in features of autodyne signal formation at distance
increase.
We should note that obtained results of theoretical studies are well agreed
with experimental data published in [1, 6]. In addition, they prejudice a cor-
rectness of explanation of the autodyne signal distortion reasons, which was
suggested in [8–10] without taking into consideration the time delay of reflected
emission.
References
1. Votoropin, S. D., Noskov, V. Ya., Smolskiy, S. M.: Modern hybrid-integrated auto-
dyne oscillators of microwave and millimeter ranges and their application. Part 2.
Theoretical and experimental investigations (in Russian). Uspehi sovremennoi ra-
dioelektroniki [Successes of modern electronic engineering], No. 7, 333 (2007)
2. Bogachev, V. M., Lysenko, V. G., Smolskiy, S. M.: Transistor oscillators and au-
todyne (in Russian). Moskva, Izdat. MEI (1993)
3. Kurokava, K.: Injection Locking of Microwave Solid-State Oscillators. Proceedings
of the IEEE. Vol. 61, No. 10, 1386–1410 (1973) https://doi.org/10.1109/PROC.
1973.9293
4. Noskov, V. Ya., Ignatkov, K. A.: Peculiarities of Noise Characteristics of Autodynes
under Strong External Feedback. Russian Physics Journal. Vol. 56, No. 12, 1445–
1460 (2013) https://doi.org/10.1007/s11182-014-0198-6
5. Noskov, V. Ya., Ignatkov, K. A.: Dynamic features of autodyne signals. Rus-
sian Physics Journal. Vol. 56, No. 4, 420–428 (2013) https://doi.org/10.1007/
s11182-013-0051-3
6. Noskov, V. Ya., Ignatkov, K. A., Smolskiy, S. M.: Autodyne Characteristic De-
pendence on the UHF Oscillators Inherent Parameters (in Russian). Radiotecnika.
No. 6, 24–42 (2012)
7. Noskov, V. Ya., Ignatkov, K. A., Smolskiy, S. M.: The role of load variations in the
formation of the autodyne response of microwave generators with a strong reflected
signal (in Russian). Radiotehnicheskie tetradi, Moskow, MEI-TU, No. 46, 33–36
(2011)
8. Lazarus, M. J., Pantoja, F. R., Somekh, M., Novak, S., Margison, S.: Nw direction-
of-motion Doppler detector. Electronics Letters. Vol. 16, No. 25, 953–954 (1980)
https:/doi.org/10.1049/el:19800679
9. Lasarus, M. J., Somekh, M. G., Novak, S., Pantoja, F. R.: Directional Harmonics
in Doppler Effect. Electronics Letters. Vol. 17, No. 2, 94–96 (1981) https://doi.
org/10.1049/el:19810068
10. Lazarus, M. J., Pantoja, F. R., Novak, S., Somekh, M. G.: Sensitivity to Direction of
Motion of a Self-Oscillating-Mixer Doppler Radar. IEE Proc. F: Commun., Radar
and Signal Processing. Vol. 129, No. 4, 233–240 (1982) https://doi.org/10.1049/
ip-f-1:19820035